In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an n-dimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the (n − k)th homology group of M, for all integers k
H k ( M ) ≅ H n − k ( M ) . {\displaystyle H^{k}(M)\cong H_{n-k}(M).}
Poincaré duality holds for any coefficient ring, so long as one has taken an orientation with respect to that coefficient ring; in particular, since every manifold has a unique orientation mod 2, Poincaré duality holds mod 2 without any assumption of orientation.
History A form of Poincaré duality was first stated, without proof, by Henri Poincaré in 1893. It was stated in terms of Betti numbers: The kth and (n − k)th Betti numbers of a closed (i.e., compact and without boundary) orientable n-manifold are equal. The cohomology concept was at that time about 40 years from being clarified. In his 1895 paper Analysis Situs, Poincaré tried to prove the theorem using topological intersection theory, which he had invented. Criticism of his work by Poul Heegaard led him to realize that his proof was seriously flawed. In the first two complements to Analysis Situs, Poincaré gave a new proof in terms of dual triangulations. Poincaré duality did not take on its modern form until the advent of cohomology in the 1930s, when Eduard Čech and Hassler Whitney invented the cup and cap products and formulated Poincaré duality in these new terms.
Modern formulation The modern statement of the Poincaré duality theorem is in terms of homology and cohomology: if M is a closed oriented n-manifold, then there is a canonically defined isomorphism H k ( M , Z ) → H n − k ( M , Z ) {\displaystyle H^{k}(M,\mathbb {Z} )\to H_{n-k}(M,\mathbb {Z} )} for any integer k. To define such an isomorphism, one chooses a fixed fundamental class [M] of M, which will exist if M {\displaystyle M} is oriented. Then the isomorphism is defined by mapping an element α ∈ H k ( M ) {\displaystyle \alpha \in H^{k}(M)} to the cap product [ M ] ⌢ α {\displaystyle [M]\frown \alpha } . Homology and cohomology groups are defined to be zero for negative degrees, so Poincaré duality in particular implies that the homology and cohomology groups of orientable closed n-manifolds are zero for degrees bigger than n. Here, homology and cohomology are integral, but the isomorphism remains valid over any coefficient ring. In the case where an oriented manifold is not compact, one has to replace homology by Borel–Moore homology
H i ( X ) → ≅ H n − i B M ( X ) , {\displaystyle H^{i}(X){\stackrel {\cong }{\to }}H_{n-i}^{BM}(X),}
or replace cohomology by cohomology with compact support
H c i ( X ) → ≅ H n − i ( X ) . {\displaystyle H_{c}^{i}(X){\stackrel {\cong }{\to }}H_{n-i}(X).}
Dual cell structures Given a triangulated manifold, there is a corresponding dual polyhedral decomposition. The dual polyhedral decomposition is a cell decomposition of the manifold such that the k-cells of the dual polyhedral decomposition are in bijective correspondence with the ( n − k {\displaystyle n-k} )-cells of the triangulation, generalizing the notion of dual polyhedra.
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